ESOLID—a system for exact boundary evaluation

We present a system, ESOLID, that performs exact boundary evaluation of low-degree curved solids in reasonable amounts of time. ESOLID performs accurate Boolean operations using exact representations and exact computations throughout. The demands of exact computation require a different set of algorithms and efficiency improvements than those found in a traditional inexact floating point based modeler. We describe the system architecture, representations, and issues in implementing the algorithms. We also describe a number of techniques that increase the efficiency of the system based on lazy evaluation, use of floating point filters, arbitrary floating point arithmetic with error bounds, and lower dimensional formulation of subproblems. ESOLID has been used for boundary evaluation of many complex solids. These include both synthetic datasets and parts of a Bradley Fighting Vehicle designed using the BRL-CAD solid modeling system. It is shown that ESOLID can correctly evaluate the boundary of solids that are very hard to compute using a fixed-precision floating point modeler. In terms of performance, it is about an order of magnitude slower as compared to a floating point boundary evaluation system on most cases

Gale duality, decoupling, parameter homotopies, and monodromy

2014 Spring.Numerical Algebraic Geometry (NAG) has recently seen significantly increased application among scientists and mathematicians as a tool that can be used to solve nonlinear systems of equations, particularly polynomial systems. With the many recent advances in the field, we can now routinely solve problems that could not have been solved even 10 years ago. We will give an introduction and overview of numerical algebraic geometry and homotopy continuation methods; discuss heuristics for preconditioning fewnomial systems, as well as provide a hybrid symbolic-numerical algorithm for computing the solutions of these types of polynomials and associated software called galeDuality; describe a software module of bertini named paramotopy that is scientific software specifically designed for large-scale parameter homotopy runs; give two examples that are parametric polynomial systems on which the aforementioned software is used; and finally describe two novel algorithms, decoupling and a heuristic that makes use of monodromy

3D Snap Rounding

Let P be a set of n polygons in R^3, each of constant complexity and with pairwise disjoint interiors. We propose a rounding algorithm that maps P to a simplicial complex Q whose vertices have integer coordinates. Every face of P is mapped to a set of faces (or edges or vertices) of Q and the mapping from P to Q can be done through a continuous motion of the faces such that (i) the L_infty Hausdorff distance between a face and its image during the motion is at most 3/2 and (ii) if two points become equal during the motion, they remain equal through the rest of the motion. In the worst case, the size of Q is O(n^{15}) and the time complexity of the algorithm is O(n^{19}) but, under reasonable hypotheses, these complexities decrease to O(n^{5}) and O(n^{6}sqrt{n})

Part of the Computer Sciences Commons Comments Victor Milenkovic & Elisha Sacks

We present two approximate Minkowski sum algorithms for planar regions bounded by line and circle segments. Both algorithms form a convolution curve, construct its arrangement, and use winding numbers to identify sum cells. The first uses the kinetic convolution and the second uses our monotonic convolution. The asymptotic running times of the exact algorithms are increased by km log m with m the number of segments in the convolution and with k the number of segment triples that are in cyclic vertical order due to approximate segment intersection. The approximate Minkowski sum is close to the exact sum of perturbation regions that are close to the input regions. We validate both algorithms on part packing tasks with industrial part shapes. The accuracy is near the floating point accuracy even after multiple iterated sums. The programs are 2% slower than direct floating point implementations of the exact algorithms. The monotonic algorithm is 42% faster than the kinetic algorithm

Planar Nef polyhedra and generic higher-dimensional geometry

We present two generic software projects that are part of the software library CGAL. The first part described the design of a geometry kernel for higher-dimensional Euclidian geometry and the interaction with application programs. We describe software structures, interface concepts, and their models that are based on cooordinate representation, number types, and memory layout. In the higher-dimensional software kernel the interaction between linear algebra and geometric objects and primitves is one important facet. In the actual design our users can replace number types, representation types, and the traits classes that inflate kernel functionality into our current application programs: higher-dimensional convex hulls and Delaunay tedrahedralisations. In the second part we present the realization of planar Nef polyhedra. The concept of Nef polyhedra subsumes all kinds of rectilinear polyhedral subdivisions and is therefore of general applicability within a geometric software library. The software is based on the theory of extended points and segments that allows us to reuse classical algorithmic solutions like plane sweep to realize binary operations of Nef polyhedra.Wir präsentieren zwei Softwareprojekte, die Teil der Softwarebibliothek CGAL sind. Der erste Teil beschreibt den Entwurf eines Geometriekerns für höherdimensionale euklidische Geometrie und dessen Interaktion mit Anwendungsprogrammen. Wir beschreiben die Softwarestruktur, die auf der Herausarbeitung von Schnittstellenkonzepten und ihren Modellen hinsichtlich Koordinatenrepräsentation, Zahlentypen und Speicherablage beruht. Dabei spielt im Höherdimensionalen die Interaktion zwischen linearer Algebra und den entsprechenden geometrischen Objekten und primitiven Operationen eine wesentliche Rolle. Unser Entwurf erlaubt das Auswechseln von Zahlentypen, Repräsentations- und Traitsklassen bei der Berechnung von d-dimensionalen konvexen Hüllen und Delaunay-Simplexzerlegungen. Im zweiten Teil stellen wir die Realisierung von planaren Nef-Polyedern vor. Das Konzept der Nef-Polyeder umfasst alle linear-polyedrisch begrenzten Unterteilungen. Wir beschreiben ein Softwaremodul das umfassende Funktionalität zur Verfügung stellt. Als theoretische Grundlage des Entwurfs dient die Theorie erweiterter Punkte und Segmente, die es uns erlaubt, vorhandene Algorithmen wie z.B. Plane-Sweep zur Realisierung binärer Operationen von Nef-Polyedern zu nutzen

Computer Science for Continuous Data:Survey, Vision, Theory, and Practice of a Computer Analysis System

Building on George Boole's work, Logic provides a rigorous foundation for the powerful tools in Computer Science that underlie nowadays ubiquitous processing of discrete data, such as strings or graphs. Concerning continuous data, already Alan Turing had applied "his" machines to formalize and study the processing of real numbers: an aspect of his oeuvre that we transform from theory to practice.The present essay surveys the state of the art and envisions the future of Computer Science for continuous data: natively, beyond brute-force discretization, based on and guided by and extending classical discrete Computer Science, as bridge between Pure and Applied Mathematics